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Question
One mapping (function) is selected at random from all the mappings of the set A = {1, 2, 3, ..., n} into itself. The probability that the mapping selected is one to one is ______.
Options
`1/n^n`
`1/n`
`(n - 1)/(n^(n - 1))`
None of these
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Solution
One mapping (function) is selected at random from all the mappings of the set A = {1, 2, 3, ..., n} into itself. The probability that the mapping selected is one to one is `(n - 1)/(n^(n - 1))`.
Explanation:
Total number of mappings from a set A having n elements onto itself is nn
Now, for one-to-one mapping the first element in A can have any of the n images in A; the 2nd element in A can have any of the remaining (n – 1) images, counting like this, the nth element in A can have only 1 image.
Therefore, the total number of one-to-one mappings is n.
Hence the required probability is `n/n^n = (n(n - 1))/(n n^(n - 1)) = (n - 1)/(n^(n - 1))`.
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